Topic outline

Lectures:When: Mondays 13:1515:00 (start 30/1)Where: Albano House 1, 3rd floor
30/1: Kovalevsky rummetfrom 6/2: Cramérrummet
Teachers: Annemarie Luger and Salvador Rodriguez Lopez
Examination: presentation and oral exam
Prequisitaries:
 Some complex analysis
 Some measure theory
 Interest in Analysis in general! 
3/4 Christian and Linus (Bergmanspaces)
17/4 usual lecture24/4 Nedialko and Ludvig (Hankel operators)
8/5 Lefteris (duality theorem)
Dario (Toeplitz operators)15/5 Ellen (Maximal function)
Vlad (Privalovs article)22/5 finishing usual lecture

 We used the pointwise majorisation result that we had for harmonic extensions of functions.
 We gave some mild conditions for convolution operators to be poitwise majorised by the HardyLittlewood Maximal function. This automatically applies to the Poisson integrals.
 We introduced the weak spaces , and the distribution function.
 We expressed the norm of a function in terms of its distribution function.
 We stated, for , the boundedness of the HardyLittlewood function, as well a the boundedness result.
 Assuming the boundedness, we proved that for . We gave an interpolation proof.
 We stated the Marcinkiewicz interpolation theorem for subadditive operators (If you revise the proof we gave for , we proved Marcinkiewicz theorem under the diagonal case and ).
 As a consequence of the general Marcinkiewiz result, we obtain Young's/Haussdorf inequality for , and announced that for , the same inequality holds provided we change by . The result for is due to Hardy and Littlewood (1927).
ReferencesZo's Lemma( More general than the one we established during the lecture) http://matwbn.icm.edu.pl/ksiazki/sm/sm55/sm5518.pdfChapter VIII (A,B1, B2)Koosis, Paul Introduction toHp spaces. Second edition. With two appendices by V. P. Havin [Viktor Petrovich Khavin]. Cambridge Tracts in Mathematics, 115. Cambridge University Press, Cambridge, 1998.Notes:1)The proof of the weak inequality uses the socalled "Rising sun Lemma". This will differ from the approach in the lectures.2) The Maximal function in the book is the socalled uncentered Maximal function, while the one in the lectures is called the centered maximal function. On can indeed show that both are pointwise equivalent to each other (Exercise).General interpolation theory:Bergh, Jöran; Löfström, Jörgen Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. SpringerVerlag, BerlinNew York, 1976.Bennett, Colin; Sharpley, Robert Interpolation of operators. Pure and Applied Mathematics, 129. Academic Press, Inc., Boston, MA, 1988. 
 We motivated the study of Maximal operators (I called the Fundamental Analysis principle [as it appears in Krantz' book]), as a natural way to study pointwise convergence. In some cases, the pointwise convergence is indeed equivalent to the boundedness of associated maximal operators (Stein's principle).
 We introduced the notion of Subharmonic Functions, and established some properties.
 We gave an equivalent characterisation, for continuous functions for being harmonic (We only proved one direction. The other can be found in the book of Duren).
 We deduced the property that averages on spheres of radius r of subharmonic, continuous nonnegative functions is nondecreasing in r.
 We stated Jensen's formula (w/o proof).
 Obtained as a corollary that given a holomorphic function in the disc then and for are subharmonic functions.
 We (finally!) give the definition of Hardy spaces .
References
Duren, Peter L. Theory of
Hp spaces. Pure and Applied Mathematics, Vol. 38 Academic Press, New YorkLondon 1970 xii+258 ppKrantz, Steven G. A panorama of harmonic analysis. Carus Mathematical Monographs, 27. Mathematical Association of America, Washington, DC, 1999. xvi+357 pp.

 We recalled the theorem regarding the existence of nontangential limits of functions in for .
 We showed that if ., those properties are preserved for functions in provided they have no zeroes.
 We proved Jensen's formula and the accompanying lemma stated in the previous lecture
 We proved the completeness of Hardy spaces a metric spaces.
 We proved that the zeroes of functions in the Nevalinna class can't be too far away from zero, so we can form a Blaschke product with them.
 That Blaschke product is an element in , with . Hence the non tangential limits exists.
 (We stated without proof) that the Blaschke product satisfies that a.e. and .
 We
stated the first factorization theorem: If or , and is not identically zero, then if is the Blaschke product
formed with the zeroes of , then we can write with and , and has no zeroes.
 We gave the proof for .
 We recalled the theorem regarding the existence of nontangential limits of functions in for .
